Answer
$$y = \left(\frac{5}{2} - \frac{1}{2}\cos t\right)^2 $$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dt}} = \sqrt y \sin t,\,\,\,\,\,\,y\left( 0 \right) = 4 \cr
& {\text{Separate the variables}} \cr
& {y^{ - 1/2}}dy = \sin tdt \cr
& {\text{Integrating}} \cr
& \frac{{{y^{1/2}}}}{{1/2}} = - \cos t + C \cr
& 2\sqrt y = - \cos t + C \cr
& {\text{Use the initial condition }}\,y\left( 0 \right) = 4 \cr
& 2\sqrt 4 = - \cos \left( 0 \right) + C \cr
& 4 + 1 = + C \cr
& C = 5 \cr
& ,{\text{ then}} \cr
& 2\sqrt y = - \cos t + 5 \cr
& {\text{Solve for }}y \cr
& \sqrt y = \frac{5}{2} - \frac{1}{2}\cos t \cr
& y = \left(\frac{5}{2} - \frac{1}{2}\cos t\right)^2 \cr} $$
